Super-friends, Sohail and Jimmy are binge-playing Super-Smash Siblings when they realize that they dont want to interact with the rest of the world for the foreseeable future (think of this as a deserted island situation caused by a mild obsession with this videogame). Their paucity of life skills means that they cannot produce anything, and they must subsist on the goods that they can find their living quarters. Sohail manages to collect 3 jars of peanut butter and 12 gummy worms, while Jimmy scrounges tougher 3 gummy worms and 5 jars of peanut butter. Although they like the same products, they have different preferences over them; Sohails utility function is (, ) = 0.5 and Jimmys utility function is (, ) = 2 2 , where denotes the number of gummy worms and denotes the number of jars of peanut butter that they consume. a. [4] Solve for the set of Pareto Optimal allocations of gummy worms and peanut butter across Sohail and Jimmy. Interpret your results. Now, suppose that the super-friends decide to engage in trade. b. [6] Normalize the price of gummy worms and find their demand functions for peanut butter. c. [2] Write the excess demand function for peanut butter. d. [6] find the equilibrium price of peanut butter ( ) that clears the two markets. How many jars of peanut butter and how many gummy worms does each of them consume in the competitive equilibrium? e. [3] Write down the competitive equilibrium. Explain the role the relative prices in ensuring that competitive markets result in a Pareto Optimal allocation. f. [4] Show that this competitive allocation is a Pareto improvement over the initial endowment.